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The problem is about finding probabilities from a normal distribution.

### Given:
- Mean (μ) = 28.6
- Standard deviation (σ) = 22.1

#### Part (a):
You need to find the probability that a single randomly selected value (X) falls between 26.3 and 28. 
This is asking for \( P(26.3 < X < 28) \).

To solve this:
1. Standardize the values using the Z-score formula:
   \[
   Z = \frac{X - \mu}{\sigma}
   \]
   - For \( X = 26.3 \), Z-score is:
     \[
     Z_1 = \frac{26.3 - 28.6}{22.1}
     \]
   - For \( X = 28 \), Z-score is:
     \[
     Z_2 = \frac{28 - 28.6}{22.1}
     \]
2. Find the cumulative probabilities associated with these Z-scores using standard normal tables or a calculator.
3. Subtract the cumulative probability of \( Z_1 \) from that of \( Z_2 \) to get the required probability.

#### Part (b):
You need to find the probability that the mean of a sample (M) of size \( n = 60 \) falls between 26.3 and 28.
This is asking for \( P(26.3 < M < 28) \).

Since you're dealing with a sample mean, the standard deviation of the sampling distribution (standard error) is:
\[
\sigma_{\text{M}} = \frac{\sigma}{\sqrt{n}}
\]
- So, the standard error is:
\[
\sigma_{\text{M}} = \frac{22.1}{\sqrt{60}}
\]
Then:
1. Use the same approach as part (a), but with the new standard deviation \( \sigma_{\text{M}} \).
2. Standardize the values using the Z-score formula, but replace \( \sigma \) with \( \sigma_{\text{M}} \).
3. Find the cumulative probabilities and subtract them as before.

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Would you like me to calculate the exact values for these probabilities?

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Here are some related questions to think about:

1. How does the standard error affect the probability in part (b)?
2. What happens to the Z-score if the sample size increases?
3. Can the normal distribution be used for small sample sizes, and why?
4. How would the results change if the standard deviation was smaller?
5. Why do we use the Z-score instead of working with the raw values directly?

### Tip:
For normal distribution problems, always convert to Z-scores to simplify working with standard normal tables.

A population of values has a normal distribution with μ = 28.6 and σ = 22.1. Find the probability that a single randomly selected value is between 26.3 and 28. Also, find the probability that a randomly selected sample of size n = 60 has a mean between 26.3 and 28.

Learn how to find the probability of a value falling between 26.3 and 28 in a normal distribution with μ = 28.6 and σ = 22.1. Explore how to calculate the probability for a sample mean with sample size 60, using the Z-score and standard error.

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